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A locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267401.png" /> whose topology is defined using a countable set of compatible norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267402.png" /> i.e. norms such that if a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267403.png" /> that is fundamental in the norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267405.png" /> converges to zero in one of these norms, then it also converges to zero in the other. The sequence of norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267406.png" /> can be replaced by a non-decreasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267408.png" />, which generates the same topology with base of neighbourhoods of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c0267409.png" />. A countably-normed space is metrizable, and its metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c02674010.png" /> can be defined by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c02674011.png" /></td> </tr></table>
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An example of a countably-normed space is the space of entire functions that are analytic in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c02674012.png" /> with the topology of uniform convergence on any closed subset of this disc and with the collection of norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026740/c02674013.png" />.
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A locally convex space  $  X $
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whose topology is defined using a countable set of compatible norms  $  \| * \| _ {1} \dots \| * \| _ {n} \dots $
 +
i.e. norms such that if a sequence  $  \{ x _ {n} \} \subset  X $
 +
that is fundamental in the norms  $  \| * \| _ {p} $
 +
and  $  \| * \| _ {q} $
 +
converges to zero in one of these norms, then it also converges to zero in the other. The sequence of norms  $  \{ \| * \| _ {n} \} $
 +
can be replaced by a non-decreasing sequence  $  \| * \| _ {p} \leq  \| * \| _ {q} $,
 +
where  $  p < q $,
 +
which generates the same topology with base of neighbourhoods of zero  $  U _ {p, \epsilon }  = \{ {x \in X } : {\| x \| _ {p} < \epsilon } \} $.
 +
A countably-normed space is metrizable, and its metric  $  \rho $
 +
can be defined by
 +
 
 +
$$
 +
\rho ( x, y)  = \
 +
\sum _ {n = 1 } ^  \infty 
 +
{
 +
\frac{1}{2  ^ {n} }
 +
}
 +
 
 +
\frac{\| x - y \| _ {n} }{1 + \| x - y \| _ {n} }
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.
 +
$$
 +
 
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An example of a countably-normed space is the space of entire functions that are analytic in the unit disc $  | z | < 1 $
 +
with the topology of uniform convergence on any closed subset of this disc and with the collection of norms $  \| x ( z) \| _ {n} = \max _ {| z | \leq  1 - 1 / n }  | x ( z) | $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,  G.E. Shilov,  "Generalized functions" , Acad. Press  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,  G.E. Shilov,  "Generalized functions" , Acad. Press  (1964)  (Translated from Russian)</TD></TR></table>

Latest revision as of 17:31, 5 June 2020


A locally convex space $ X $ whose topology is defined using a countable set of compatible norms $ \| * \| _ {1} \dots \| * \| _ {n} \dots $ i.e. norms such that if a sequence $ \{ x _ {n} \} \subset X $ that is fundamental in the norms $ \| * \| _ {p} $ and $ \| * \| _ {q} $ converges to zero in one of these norms, then it also converges to zero in the other. The sequence of norms $ \{ \| * \| _ {n} \} $ can be replaced by a non-decreasing sequence $ \| * \| _ {p} \leq \| * \| _ {q} $, where $ p < q $, which generates the same topology with base of neighbourhoods of zero $ U _ {p, \epsilon } = \{ {x \in X } : {\| x \| _ {p} < \epsilon } \} $. A countably-normed space is metrizable, and its metric $ \rho $ can be defined by

$$ \rho ( x, y) = \ \sum _ {n = 1 } ^ \infty { \frac{1}{2 ^ {n} } } \frac{\| x - y \| _ {n} }{1 + \| x - y \| _ {n} } . $$

An example of a countably-normed space is the space of entire functions that are analytic in the unit disc $ | z | < 1 $ with the topology of uniform convergence on any closed subset of this disc and with the collection of norms $ \| x ( z) \| _ {n} = \max _ {| z | \leq 1 - 1 / n } | x ( z) | $.

References

[1] I.M. Gel'fand, G.E. Shilov, "Generalized functions" , Acad. Press (1964) (Translated from Russian)
How to Cite This Entry:
Countably-normed space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Countably-normed_space&oldid=12036
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article